Definition of Powers in Stromberg's Introduction to Classical Real Analysis

Question

On pages 136-137 of his book Introduction to Classical Real Analysis Stromberg makes the following definition:

Am having qualms about his definition $0^b=0$ for any complex number $b\neq 0$, which doesn't seem to agree with usual conventions (e.g. $0^{-1}$ is usually undefined). Is this a mistake? Or is there a reason for making such an extended definition?

Any help is greatly appreciated.

Answer 1

You (and Calum Gilhooley) are right to be dubious about this. For $z=x+\mathrm iy\in\Bbb C$, with $x,y\in\Bbb R$, it is problematic to assign any meaning to $0^z$ unless $x>0$ or $x=y=0$. It would be interesting to know whether Stromberg actually makes use of his “definition” of $0^z$ in the cases $x<0$ or $x=0$ with $y\neq0$.